Question

A hollow steel shaft with and outside diameter of (do)-420 mm and an inside diameter of (di) 350 mm is subjected to a torque of 300 KNm, as shown. The modulus of rigidity G for the steel is 80 GPa. Determine: (a) The maximum shearing stress in the shaft. (b) The shearing stress on a traverse cross section at the inside surface of the shaft (c) The magnitude of the angle of twist for a (L) -2.5 m length.

Answer 1

Answer:

a.  $\tau=51.55 MPa$

b.$\tau=42.95MPa$

c.$\theta=7.67\times 10^{-3}$ rad.

Explanation:

Given: $D_i=350 mm,D_o=420 mm,T=300 KN-m ,G=80 G Pa$

We know that

$\dfrac{\tau}{J}=\dfrac{T}{r}=\dfrac{G\theta}{L}$

J for hollow shaft $J=\dfrac{\pi (D_o^4-D_i^4)}{64}$

(a)

 Maximum shear stress $\tau =\dfrac{16T}{\pi Do^3(1-K^4)}$

      $K=\dfrac{D_i}{D_o}$⇒K=0.83

$\tau =\dfrac{16\times 300\times 1000}{\pi\times 0.42^3(1-.88^4)}$

   $\tau=51.55 MPa$

(b)

We know that $\tau \alpha r$

So $\dfrac{\tau_{max}}{\tau}=\dfrac{R_o}{r}$

$\dfrac{51.55}{\tau}=\dfrac{210}{175}$

$\tau=42.95MPa$

(c)

$\dfrac{\tau_{max}}{R_{max}}=\dfrac{G\theta }{L}$

$\dfrac{51.55}{210}=\dfrac{80\times 10^3\theta }{2500}$

$\theta=7.67\times 10^{-3}$ rad.